1. Field of the Invention
The present invention relates to simulator apparatus for a combined simulation of electromagnetic wave analysis and circuit analysis, as well as to computer-readable media containing a simulation program designed therefor. More particularly, the present invention relates to a simulator apparatus for a combined simulation of electromagnetic wave analysis and circuit analysis, in which the time step size of the circuit analysis may vary in accordance with the circumstances, and a computer-readable medium containing a simulation program that performs such a simulation.
2. Description of the Related Art
In today's computational electromagnetics, the Finite-Difference Time-Domain (FD-TD) method is used as a practical technique to analyze the transitional behavior of electromagnetic waves by using a computer for numerical simulation. The FD-TD, which solves Maxwell's equations in the time and spatial domains with difference methods, is actually implemented to a variety of cases because of its wide scope of applications.
In the field of circuit analysis, on the other hand, there are different computer simulation tools which numerically analyze the transient behavior of an electronic circuit. Generally, they calculate the circuit's node voltages and branch currents on the basis of Kirchhoff's laws. The circuit analysis tools are further equipped with programs for modeling the characteristic equations of semiconductor devices and other circuit elements having nonlinear voltage-current characteristics, and to solve those nonlinear equations, they employ some numerical techniques such as Newton-Raphson methods.
Researchers have also proposed another numerical simulation method that performs both an electromagnetic wave analysis and a circuit analysis in a combined manner. In this proposed method, the simulation steps proceed along the time axis, keeping a close linkage between the electromagnetic field physics obtained by the electromagnetic wave analysis and the voltage-current physics obtained by the circuit analysis. With capabilities of both an electromagnetic wave analysis and a circuit analysis, this hybrid numerical simulation technique provides an advantage in that it solves the characteristics of circuit elements and the electromagnetic field behavior in their surrounding space in a unified manner. It is particularly useful for the analysis of high-frequency signals propagating in a circuit.
When using FD-TD techniques to numerically simulate the transient behavior of an electromagnetic wave, it is desirable that the time step size (.DELTA.t.sub.em) be as large as possible within a range determined by the following condition for stability. ##EQU1##
Here, v is the speed of electromagnetic waves; .DELTA.x.sub.min, .DELTA.y.sub.min, and .DELTA.z.sub.min represent the minimum values of spatial discretization intervals defined in the x-, y-, and z-directions, respectively. The above constraint about the time step size derives from the following reasons: (a) the numerical solution would diverge if .DELTA.t.sub.em was larger than the right side of the condition (1), and (b) the overall computation time of numerical simulation would increase if an unnecessarily small value was set to .DELTA.t.sub.em. Usually, the time step size .DELTA.t.sub.em is determined within the following range. ##EQU2##
Once it is determined, the value of .DELTA.t.sub.em is used without changes, throughout the simulation period.
As opposed to this, the time step size (.DELTA.t.sub.cs) in the transient circuit simulation normally varies in accordance with the circumstances. For example, when numerically solving nonlinear circuit equations, the time step size should be reduced if the solution for potential differences does not converge. To allow the electromagnetic wave analysis and circuit analysis to exchange data with each other, it is therefore necessary to synchronize their simulation clocks in a reliable and efficient manner, so that they will transfer data to each other at the definite time points.
Now, the next paragraphs will discuss the timing of data transfer operations between the electromagnetic wave analysis and the circuit analysis.
FIG. 9 is a diagram which shows the timing of data transfer in a conventional simulator. The upper half of FIG. 9 shows a process of electromagnetic wave analysis, while the lower half shows a process of circuit analysis. In the equations appearing in the following section, the superscripts (e.g., n, n+1/2, . . . ) affixed to the variables indicate a specific time step k, or the k-th time step within a simulation process, meaning that a time of (k.times..DELTA.t.sub.em) has elapsed since the beginning of the process.
(S31) The simulator calculates magnetic field values at t.sub.em =(n+1/2).DELTA.t.sub.em, where n is a natural number, and t.sub.em is a simulation time in the electromagnetic wave analysis. Here, the magnetic field H is calculated by EQU H.sup.n+1/2 =H.sup.n-1/2 -(.DELTA.t.sub.em /.mu.) rot E.sup.n (3)
where .mu. is permeability, and "rot" represents the rotation (or curl) of a vector function.
(S32) The simulator then calculates current source data I on the basis of the magnetic field H obtained in step S31 as part of the electromagnetic wave analysis, and transfers the resultant value to the circuit analysis. This current source data I is obtained by the following. EQU I.sup.n+1/2 =.SIGMA.H.sup.n+1/2 (4)
(S33) Based on the current source data I supplied from the electromagnetic wave analysis, the simulator executes a circuit analysis at t.sub.cs =(n+1/2).DELTA.t.sub.em, where t.sub.cs is a simulation time in the circuit analysis. Here, the behavior of the circuit is given by the following equation. EQU V.sup.n+1/2 =Z.sup.-1 I.sup.n+1/2 (5)
(S34) Based on the voltage V obtained through the circuit analysis, the simulator calculates the electric field E in a specific part (i.e., cell) of the computational domain in which the circuit is located. The electric field E, given by the following, is then passed to the electromagnetic wave analysis.
E.sup.n+1 =V.sup.n+1/2 /d, (6)
where d is the dimension of a cell in which the circuit under analysis is located.
(S35) The simulator calculates the electric field at t.sub.em =(n+1).DELTA.t.sub.em, according to the following equation. EQU E.sup.n+1 =E.sup.n +(.DELTA.t.sub.em /.epsilon.) rot H.sup.n+1/2, (7)
where .epsilon. is a dielectric constant. The electric field data supplied in step S34 is then adapted to the electric field calculated in step S35, thereby yielding the entire electric field at t.sub.em =(n+1).DELTA.t.sub.em.
The combined simulation for electromagnetic wave analysis and circuit analysis is accomplished in this way, by allowing the two processes to exchange the current source data and electric field data. The conventional simulation process, however, is likely to produce an unstable solution even if the time step size .DELTA.t.sub.em satisfies the stability condition (2). This problem seems to be caused by a time lag of .DELTA.t.sub.em /2 between the magnetic field calculation (S31) and the electric field calculation (S35) as part the electromagnetic wave analysis. The conventional simulation process, however, does not take that time lag into consideration, unfortunately. That is, the simulator calculates the electric field at a specific time point (n+1).DELTA.t.sub.em on the basis of the voltage at a past time point (n+1/2).DELTA.t.sub.em, which makes the solution unstable. It may be possible to avoid such instability of solutions by reducing .DELTA.t.sub.em to about half the standard step size that is suggested by the stability condition (2). This choice, however, leads to two times longer computation time. Further, the value of t.sub.cs is varied in accordance with the progress of the circuit analysis, and therefore, ten is not always equal to t.sub.cs. In a conventional simulation process, the analysis often fails due to the inconsistency in t.sub.cs and t.sub.em.